A Harmonic Balance Approach for Bifurcation

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SAAEI 2014. Tangier, 25-27 June. CDES-6
A Harmonic Balance Approach for Bifurcation
Analysis of a Ferroresonant Circuit
J. A. COREA-ARAUJO, A. EL AROUDI, F. GONZÁLEZ-MOLINA, J. A. MARTÍNEZ-VELASCO, J. A. BARRADO-RODRIGO, AND
L. GUASCH-PESQUER
Abstract—Despite the extensive available literature on this
topic, ferroresonance prediction and behavior characterization
still remain widely unknown. The Harmonic Balance Method is
presented in this paper as an approach to predict the dynamics of
a ferroresonance system. The implication of initial conditions in
the final ferroresonant state is analyzed. Bifurcation diagrams
are obtained to depict the path in which the phenomenon will
move and to predict the critical points where ferroresonance
occurs. Stability region for the test system is also located from the
study.
Keywords: Ferroresonance, Harmonic Balance, Bifurcation,
Stability, Transformers.
F
I. INTRODUCTION
ERRORESONANCE is one of the most common transient
and steady-state phenomena studied by power system
engineers. The phenomenon usually appears in scenarios
involving a non-linear inductance associated with any
capacitance source. Such interaction unfolds not only into
potentially dangerous overvoltages, jumping to a high current
fundamental-frequency state but also bifurcations to subharmonic, quasi-periodic and even chaotic oscillations. The
phenomenon is commonly initiated after some type of
switching event such as load rejection, fault clearing,
transformer energization, single-phase switching or loss of
system grounding. In general, ferroresonance can occur in any
power system that contains transformers, regardless its type or
size. Capacitance interaction can be found either in the form of
actual capacitor banks or capacitive coupling [1].
Although ferroresonance has been present in power systems
for more than a century, and despite the extensive literature
available nowadays, its behavior and characterization still
remain widely unknown [2]. The first analytical works were
presented in the 50’s [3], [4]. In the 70’s and 80’s, digital
simulation began to replace transient network analyzers
(TNA) following pioneering work in digital computer solution
methods by Dommel [5].
After the breakthrough of the nonlinear dynamics and chaos
theory in the late 70’s, a new analysis door was open. The
Javier A. Corea-Araujo, Abdelali El Aroudi, Francisco González-Molina,
José A. Barrado-Rodrigo, and Luis Guasch-Pesquer are with the Departament
d'Enginyeria Electrònica, Elèctrica i Automàtica, Universitat Rovira i Virgili,
Av. Països Catalans 26, 43007 Tarragona, Spain (e-mail: [email protected];
[email protected];
[email protected]; [email protected])
Juan A. Martinez is with the Departament d'Enginyeria Elèctrica,
Universitat Politècnica de Catalunya, Av. Diagonal 647, 08028 Barcelona,
Spain. (e-mail: [email protected])
connection of ferroresonance to nonlinear dynamics and chaos
allowed to increase the expertise in waveform analysis and
enabled a classification of ferroresonance states into different
modes [6]: Fundamental mode (voltages and currents are
periodic with a period T equal to the power frequency period),
sub-harmonic mode (periodic signals with a period multiple of
the source period), quasi-periodic mode (non-periodic signal,
composed by two or more periodic signals with incommensurable period), and chaotic mode (signals showing an
irregular and unpredictable behavior).
The first work linking up ferroresonance with non-linear
analysis was published in 1992 [7]. That work presented a
three-phase transformer leading to a chaotic state. Topics such
as bifurcation theory, global dynamic behavior or the Galerkin
method were common for ferroresonance analysis during the
90’s [8],[9].
However, it was not until 2000 when several techniques
obtained from the merge with nonlinear dynamics were
developed [10]. The first significant contribution was
presented in [11]. Research topics such as ferroresonance
behavioral patterns [12], hysteresis impacts [13], transformer
core studies [14] or new transformer models derivation [15],
are usually followed by the words ‘stability domain’,
‘dynamic behavior’ or ‘harmonic balance’ [16].
In this research, a predictive analysis is presented to locate
stability domains for a period-1 ferroresonance. This
prediction is made by using the Harmonic Balance method to
solve the differential equation of the system. The order of
polynomial expression representing the non linearity has been
strategically selected to approach the physical characteristic of
the inductor. The solution technique used in the harmonic
balance allows comparing the procedure with the widely
studied Duffing’s equation solution [11].
The document is organized as follows. A detailed
description of the system under study is presented in Section
II. The implementation of the Harmonic Balance Method is
presented in Section III. Validations through bifurcation
diagrams obtained from system level simulations are presented
in Section IV. Conclusions are given in Section V.
II. TEST SYSTEM DESCRIPTION
To better explain the approach of this paper, the basic
circuit shown in Fig. 1, will be analyzed. The circuit is
composed of a 25 kV/50 Hz power source, a capacitance of 5
µF and a saturable inductance paralleled with a resistance of
40 kΩ representing the core losses [1].
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III. THE HARMONIC BALANCE METHOD
Fig. 1. Diagram of the test circuit
A proper representation of the transformer saturation is
considered vital in ferroresonance studies. Hence, when a
mathematical equivalence of a transformer circuit is
developed, the expression representing the saturation effect
must be close-fitted to actual test data. Several approaches
have been used throughout the years, being the two most
accurate the trigonometric approximation based in the
hyperbolic arctangent properties [17], and the polynomial
approximation, whose strength relies on its simple
implementation and the easy calculation of its coefficients
[11], [18].
In this work, the polynomial approximation has been used
due to its suitability on the harmonic balance resolution and
the possibility of comparing with previous works. Expression
(1) shows the polynomial form.
(1)
where iL represents the magnetizing current, λ represents the
fluxlinked and a and b the coefficients for the curve fitting.
Selecting a proper value of n in the polynomial expression will
significantly mark the accuracy of representation. Figure 2
shows different curves corresponding to (1) with different n
values and the actual test data. The figure proves that higher
values of n approach better to the real nature of magnetization
curve and its utilization will make the modeling more realistic.
From Figure 2, the differential equation representing the
mathematical behavior of the system is as followed:
(2)
cos
where k= 1/RmCs, C1= a/Cs, C2= b/Cs and G=ωVrms. In order
to best approach the physical reality of the saturation curve ‘n’
is set to 11. The coefficients are set as follows:
43.59
10 and
7.415 10 .
The harmonic balance is a method for the study of nonlinear oscillating systems which are defined by non-linear
ordinary differential equations [19].
Few approaches has been made in the ferroresonance field,
most of them only consider the sinousoidal approximation or
consider a low order polynomial approximation [20], [21]. In
general terms, the method consists on substituting the
unknown variable in the equation by an assumed solution so
that approximate periodic solutions can be found. The forced
solution can be selected to behave as a steady state solution
using a truncated Fourier series [18].
#
*
$ %& cos
'& sin
+,#
(3)
In this work the dc component can be neglected due to the
sinusoidal state of the source signal. Here k sets the number of
harmonic components. Normally this value depends on the
oscillations considered or expected. For this work, k is set to 1
to obtain a first order approximation. Thus, the equation (3)
will have the form:
(4)
% cos
' sin
In order to substitute (4) into (2), the first and second
derivatives should be found. These are:
- % sin
' ωcos
(5)
(6)
-% ω cos
- ' ω sin
Substituting the expression for the flux and its derivatives in
(2) and having the value ‘n’ as 11, will significantly change
the expression (2) in size and complexity. Once simplification
is performed, and the third and higher harmonics are
neglected, a set of expressions are obtained for first-order
solution of the parameters A and B. These are given as
follows:
%/
-
'/
-
231
512
%
'
1
%
'
1
231
512
2
'
(7)
2-%
0
(8)
A general expression can be found by using the polar
solution of the flux:
where
% cos
' sin
3 cos
4
(9)
(10)
3 5%
'
Thus, r by convenience provides also the magnitude of the
flux. Then, (7) and (8) merges in a single expression:
6 3
Fig. 2. Non-linear inductance characteristic
3 /
-
231
512
3 #2
(11)
According to [9] real values of r that satisfy (11) correspond
to the periodic solutions for the original differential equation
(2). A graphical explanation of (11) can be made using a 1-D
bifurcation diagram. It is obtained by founding the analytically
solution for (2) while a system parameter is varied (source
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voltage). Figure 2 shows the result for the system. It can be
deduced from this figure that, when the bifurcation parameter
increases, the response of the flux suddenly jumps when it
approaches the first turning point (TP1). The same will happen
when the bifurcation parameter is slowly decreased; the flux
will abruptly drop while approaching the second turning point
(TP2). This type of behavior is usually referred as hysteretictype pattern and it is associated to a saddle-node bifurcation
[9]. The frontier at which the jumps occur can be determined
by founding the threshold of the change just when (10) has
mind that the harmonic balance prediction is based on a firstorder harmonic approach, which implies that only the
sinusoidal stage is considered whereas computational systems
include all the harmonic series in their solution. In addition, it
is important remarking that EMTP-ATP simulation can take
several hours without including the post-processing task.
Harmonic balance can take no more than an hour.
78
only one solution instead of two. This occurs when
0.
79
The derivative of (11) will give solution for (2) at the turning
points and equating it to 0.
23 :
11
=
3
-
-
;
;
1
1
3
#
3
#
<
0
(12)
If different capacitance values are tested, the corresponding
values of voltage for the turning points can be calculated
evaluating (13) for Vrms, and using the pairs (r,Cs) obtained
from (12) [13].
3 /
-
231
512
3 #2
>[email protected]
(13)
The graphical representation of equation (13) is known as
bifurcation line or stability domain boundary [11]. The 2-D
bifurcation map from Figure 3 has a significant meaning for
ferroresonance analysis: it allows predicting where the
boundary between sinusoidal and ferroresonance response is
located. For instance, every combination of values at the right
of TP1 line will have a high flux peak value while left side
combinations could have both low flux values for TP1
conditions and high flux values response due to the
coexistence with TP2 conditions.
Fig. 2. 1-D bifurcation diagram for Cs=5µF.
IV. BIFURCATION DIAGRAMS FROM THE COMPUTATIONAL
SYSTEM MODEL
In order to benchmark the analysis, a set of bifurcation
maps will be presented from the test system. Time domain
simulations will be carried out under MATLAB environment
to validate the mathematical analysis and the veracity in the
harmonic balance approximation.
Figure 4 shows the bifurcation diagram obtained from time
domain simulation. This result proves the two different
solutions predicted by the harmonic balance. One obtained by
parametrically increasing the voltage, and the other by
decreasing the same parameter. It is important to emphasize
the co-existence of the two solutions, which reinforces the
influence that initial conditions have over the behavior of a
ferroresonant system.
To confirm the result derived from the harmonic balance
analysis, an EMTP simulation study has been performed.
Figure 5 presents the 1-D bifurcation diagram originated from
EMTP-ATP simulation. The results are significantly close to
harmonic balance prediction. A small discrepancy can be
detected on the upper stable solution when comparing Figure 2
against Figures 4 and 5. However, it is important to keep in
Fig. 3. 2-D bifurcation diagram.
Fig. 4. 1-D bifurcation diagram obtained using MATLAB for
Cs=5µF.
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V. CONCLUSIONS
Fig. 5. 1-D bifurcation diagram obtained using EMTP-ATP for
Cs=5µF.
This paper has presented the application of the Harmonic
Balance Method for ferroresonance analysis of a simplified
transformer model by respecting the true nature of the
magnetic saturation. The solutions were derived from a highorder polynomial so the results are close to the physical reality
of a transformer. Comparisons of time-domain simulations
using MATLAB and EMTP-ATP were presented. The
analysis presented in this work has remarked the influence of
the initial conditions in ferroresonance representation; the
proper characterization can be vital for a well developed
analysis.
The main contribution of the paper has been the application
of harmonic balance as a predictive tool. Its capability to
describe the behavior tendency of a system prone to
ferroresonance makes it a powerful technique. At the same
time its simplicity makes it suitable to substitute large
parametric analysis avoiding extensive time-domain
simulations. Finally, bifurcation maps resulting from harmonic
balance analysis provides clear paths for system design that
could avoid hypothetical ferroresonance conditions.
VI. ACKNOWLEDGMENT
This work is being supported by the Spanish “Ministerio de
Educación y Ciencia” under the Grant DPI2012-31580.
VII. REFERENCES
Fig. 6. System response for Vrms=25 kV and Cs=10 µF.
Fig. 7. System response for Vrms=9 kV and Cs=10 µF .
From Figure 3 it is easy to differentiate the regions for the
stability of the systems. Any match of parameters selected at
the right of the TP1 boundary will result into an undesirable
solution; that is, in a ferroresonant state. A time-domain
simulation has been performed to prove the previous
statement. Figure 6 validates the prediction for the undesirable
region. However, the region at the left of TP1 in Figure 3 is
totally dependent of the initial condition on the system. Two
different stages could occur. On one hand, the system could
work in a sinusoidal state, normally for initial conditions set to
zero. On the other hand, due to initial conditions (i.e.,
remanent flux and initial capacitance voltage), the response of
the system could be unstable; in other words, ferroresonant.
Figure 7 shows the co-existence of the two response signals
for the same set of parameters.
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